Dedekind-Hasse norm implies principal ideal ring
Statement
A commutative unital ring that admits a Dedekind-Hasse norm must be a principal ideal ring.
In particular, an integral domain that admits a Dedekind-Hasse norm must be a principal ideal domain.
Definitions used
Dedekind-Hasse norm
Further information: Dedekind-Hasse norm
A Dedekind-Hasse norm on a commutative unital ring is a function
from the nonzero elements of
to the nonnegative integers with the property that for any elements
with
, it is true that either
or there exists an element
in the ideal
such that
.
Principal ideal ring
Further information: Principal ideal ring
A commutative unital ring is termed a principal ideal ring if every ideal of
is principal.
Proof
Given: A commutative unital ring with a Dedekind-Hasse norm
. An ideal
of
.
To prove: There exists such that
.
Proof: If , we can take
, and we are done.
So, suppose . Consider the function
on the nonzero elements of
. Since
maps to a well-ordered set, there is an element
such that
for all
.
Pick any . If
, then
or there exists
with
.
Note that in the latter case, we have , so
, and
, contradicting the assumption that
has minimum norm among the nonzero elements of
. Hence, we have the case
. Thus,
. Conversely, we clearly have
, so
.